Both orbitals look like $\pi$ orbitals, as both have nodes at the plane of the molecule, which is characteristic for $\pi$ systems. Occupied valence orbitals are usually considered to be bonding while excited unoccupied orbitals are often antibonding. In this particular case I woudn't say that the distinction between bonding and antibonding is 100% clear, but I'd call the LUMO $\pi^*$ in this case since the nodal structure can clearly be seen. I would also call this a charge transfer transition since you have a clear transfer of density from the left benzodioxole group to the central indene(?) group and the right benzene group.
But orbital characterization is not clear cut process. As you know yourself, there is no clear definition of these terms except in the most basic systems and I would recommend not to overthink the characterization. Just make sure that you you don't go against already established characterizations in literature.
I would have preferred to post this as a comment as I don't provide a general methodology to solve the characterisation problem. But my comment didn't fit in the comments field.